Introduced here is the STEPHANOID CURVE[ authors proposal of nomination] , the mirror images of two Archimedes curves of a special form,passing Zero x-y point and on which lie vector length logarithms, of vector line lengths on the spiral[ one on top of the other].

Let X=e^[ Θ/90], be a point on the curve of this Spiral of Logarithmic base e ,on the plane of x-y co-ordinate system of Cartesian axes.Phasor OX is at angle Θ from the x-axis taking anticlockwise direction as positive(Θ on x-axis is taken zero )and clockwise negative.This Spiral crosses the [-y ] axis at point [ 1/e ],then the x axis at point 1 [moving anticlockwise] ,following crosses the y axis at point with value e ,then the [-x] axis at e^2... etc...This gives ln(X)=[ Θ/90], OR in RADIANS ln(X)= Θ/[Pi/2].

The 0 of the x y cross section is the assymptotic point of the Spiral.

e^x = 1+x+(x^2/2!)+(x^3/3!)+(x^4/4!)+(x^5/5!)may be transformed to other than e bases.For instance,Lets for reasons that serve my web work(references below) , instead of x use Zwhere Z is a ral positive or negative number.

So e^Z= 1+ Z+ [Z^2]/2!+…+.....

Let X=e^Z , AND Log_e[X]=Z

So X= 1 + Log_e[X] + {[Log_e[X] ]^2}/2!+....

Raise e^Z to the power of N (POSITIVE REAL NUMBER),we get

e^[NZ]=1+NZ+{[NZ]^2}/2!+...

Let e^N = T SO THAT Log_e[T]=N

WHERE T is a POSITIVE REAL NUMBER[BASE OF ANOTHER THAN e LOGARITHM ] , then ,

Powers of a Length A[ab] Graphically by Compass and Ruler.[The theory simply lies on the "similar orthogonal triangles"]

Draw a line horizontally of the Unit length of the Ruler.Let it be [ej] ,and A avertical line [jh] ,at j corresponding to the Length A ,picked up by the compass.Then [eh] will be the hypotenuse of the orthogonal triangle [ejh].Let Θ be the angle whose tangent is: [hj]/[ej]=A/1=A.We extend the line [ej] to [k],and [eh] to [q].We use the compass and place the length A on [ek] ,so we get a point on it [m] , that [em]=A.Then , we draw a vertical line on [em] at [m],which meetsthe line [eq] at [n].Then [mn]=[A^2] , and tan(Θ) = A.So we have graphically obtained the square of the Length A , GRAPHICALLY BY COMPASS and RULER .We pick up the length [A^2] lay it along line [ek] sowe get a point on it [r] ,so that [er]=A^2 , the verticalon it ,meets [eq] at [s] ,so [rs]=A^3, and tan(Θ)=A.By doing so we get the powers we need of a length graphically by compass and ruler.So any series that involves powers of a length x[ such as e^x=1+x+[1/2!]x^2+... , log x , trigs etc ], may be drawn graphically.Of course a similar method is involved for the productof two lengths , inverses ,etc.[The inverse 1/A is obtainedfrom the original triangle taking as vertical the Unit Length of the ruler.Then the horizontal length is 1/A and tan(Θ)=A]

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